Stokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars
aa r X i v : . [ a s t r o - ph . S R ] A p r Mon. Not. R. Astron. Soc. , 1–26 (2011) Printed 12 November 2018 (MN L A TEX style file v2.2)
Stokes tomography of radio pulsar magnetospheres. II.Millisecond pulsars
C. T. Y. Chung ⋆ and A. Melatos School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
ABSTRACT
The radio polarization characteristics of millisecond pulsars (MSPs) differ significantlyfrom those of non-recycled pulsars. In particular, the position angle (PA) swings ofmany MSPs deviate from the S-shape predicted by the rotating vector model, even af-ter relativistic aberration is accounted for, indicating that they have non-dipolar mag-netic geometries, likely due to a history of accretion. Stokes tomography uses phaseportraits of the Stokes parameters as a diagnostic tool to infer a pulsar’s magneticgeometry and orientation. This paper applies Stokes tomography to MSPs, generaliz-ing the technique to handle interpulse emission. We present an atlas of look-up tablesfor the Stokes phase portraits and PA swings of MSPs with current-modified dipolefields, filled core and hollow cone beams, and two empirical linear polarization models.We compare our look-up tables to data from 15 MSPs and find that the Stokes phaseportraits for a current-modified dipole approximately match several MSPs whose PAswings are flat or irregular and cannot be reconciled with the standard axisymmetricrotating vector model. PSR J1939+2134 and PSR J0437 − α, i ) = (22 ± ◦ , ± ◦ ) and emission altitude 0.4 r LC . The fit is less accuratefor PSR J1939+2134 at 1.414 GHz, and for PSR J0437 − Key words: magnetic fields — polarization — pulsars: general — pulsars: individual:PSR J0437 − The two tools used most frequently to characterise the orien-tation and magnetic geometry of a radio pulsar are its pulseprofile and position angle (PA) swing. The rotating vec-tor model (Radhakrishnan & Cooke 1969), which assumesan axisymmetric magnetic field, predicts an S-shaped swingacross one pulse period and is traditionally used to deter-mine the inclinations of the magnetic axis of symmetry andthe observer’s line of sight to the rotation axis. However, lim-itations arise when analysing only the PA swing, especiallyas the magnetosphere is not axisymmetric in general, e.g.the magnetic field includes a current-modified component(Hibschman & Arons 2001).In Chung & Melatos (2010) (hereafter CM10), Stokestomography was introduced as a diagnostic tool to be usedalongside more traditional methods of analysis. It exploitsthe fact that the phase portraits traced out by the fourStokes parameters, when plotted against each other over ⋆ E-mail: [email protected] one pulse period, are unique for any given magnetic ge-ometry and orientation. An atlas of look-up tables, con-taining Stokes phase portraits and PA swings, was gener-ated by CM10 for a variety of simple models, includingpure and current-modified dipole fields, filled core and hol-low cone beams, and the associated linear polarization pat-terns. CM10 also showed that, from a sample of 24 nominally“dipolar” pulsars, which obey the period-pulse-width rela-tion and/or exhibit clean S-shaped PA swings, the Stokesphase portraits of 16 objects are either inconsistent withlow-altitude emission from a pure dipole field, or have ahighly asymmetric surface emission pattern.In this paper, we turn our attention to millisecondpulsars (MSPs). Polarimetric studies of MSP radio emis-sion have uncovered complex behaviour not normally seenin slower pulsars. In particular, the PA swings of manyMSPs are neither clean nor S-shaped; instead, they are flat(Stairs et al. 1999; Ord et al. 2004), highly distorted (e.g.PSR J0437 − − c (cid:13) C. T. Y. Chung et al. ing that the magnetic geometry changes a lot with altitude,or that the observed pulse profile comprises emission fromseveral different regions and altitudes.The above trends suggest that MSPs have nondipo-lar magnetic fields. In a non-recycled pulsar, a dipole fieldcan be distorted by several mechanisms, e.g. a currentflowing along the field lines (Hibschman & Arons 2001;Dyks 2008), or rotational sweepback near the light cylin-der (Hibschman & Arons 2001; Dyks & Harding 2004; Dyks2008). In a recycled pulsar with a history of prolonged ac-cretion, another set of mechanisms comes into play. For ex-ample, accreted material channeled onto the magnetic polesdistorts the frozen-in magnetic field as it spreads towardsthe equator (Melatos & Phinney 2001; Payne & Melatos2004; Zhang & Kojima 2006; Vigelius & Melatos 2008).Quadrupolar magnetic fields, proposed to explain the X-ray light curves of Her X-1 (Shakura et al. 1991), can evenbe comparable to the dipolar component (Long et al. 2008).Multipole fields can also be generated near the inner edgeof the partially diamagnetic accretion disk of an X-ray pul-sar (Lai et al. 1999). Alternatively, as the pulsar is spun upby accretion, the magnetic pole drifts towards the rotationaxis, dragged inward by the motion of superfluid vorticesin the pulsar’s core (Srinivasan et al. 1990; Ruderman 1991;Cheng & Dai 1997; Lamb et al. 2009).In this paper, we apply Stokes tomography to millisec-ond pulsar data drawn from the European Pulsar Network’s(EPN) online database. In Section 2, we briefly review thefitting recipe for determining the optimal orientation andbeam polarization patterns from observed pulse profiles andStokes phase portraits. We also extend the model in CM10 totreat interpulse emission. We compare our improved look-uptables of Stokes phase portraits and PA swings to observa-tions of 15 MSPs in Section 3 to identify general trends. Wethen conduct detailed modelling of PSR J1939+2134, whichhas a strong interpulse, and PSR J0437 − For the convenience of the reader, we begin by summarisingbriefly how to determine the emission point and hence thepolarization state of the radiation as a function of pulselongitude, following the recipe laid out in Section 2 of CM10.Our notation and definitions copy CM10.We define two reference frames, as in Figure 1 of CM10:the inertial frame, in which the observer is at rest, with axes( e x , e y , e z ), and the body frame of the pulsar. The relativemotion between the frames is computed by solving Euler’sequations of motion (including precession in general but notin this paper). The line-of-sight vector w is chosen to lie inthe e y - e z plane, making an angle i with e z . The rotationand magnetic axes lie along e z and one of the body frameaxes ( e ) respectively, separated by an angle α . We definea spherical polar grid ( r, θ, φ ) in the body frame coveringthe region x min r/r LC x max , 0 θ π , 0 φ π , with 64 × ×
128 grid cells, where the line θ = 0lies along e , and r LC = c/ Ω is the light cylinder radius. In this paper, we take x min = 0 .
01 and x max = 0 .
83 toaccomodate the relatively small magnetospheres (and henceemission altitudes) of MSPs.Radiation from highly relativistic particles flowingalong magnetospheric field lines is narrowly beamed. Hence,without relativistic aberration, the observed emission point x ( t ) at any time t is located where the magnetic vector B [ x ( t ) , t ] points along w . When aberration is included,the emission point x ( t ) at time t satisfies the equation(Blaskiewicz et al. 1991; Dyks 2008) w = ± t + Ω × x /c |± t + Ω × x /c | , (1)where t = B [ x ( t ) , t ] / | B [ x ( t ) , t ] | is the unit tangent vectorto the magnetic field at x ( t ), and Ω is the angular veloc-ity vector. CM10 considered emission from only one pole forsimplicity [i.e. + t in equation (1)], and hence ignored in-terpulse emission. In this paper, we include emission fromboth the north and south poles, requiring both ± t termsto be retained in (1). At every instant, we thus have fouremission points which satisfy (1), two in the hemisphere op-posite the observer (which he cannot see), and two facingthe observer, which we label P and P . We search the gridat a fixed altitude r to find P and P ; the locations of P and P change with time in both the body frame and the in-ertial frame. The Ω × x term in (1) encodes the aberrationeffect, as in Hibschman & Arons (2001). It is correct to or-der O ( r/r LC ) and should be replaced by the full relativisticexpression when warranted by confidence in the model anddata.The Stokes parameters ( I, Q, U, V ) associated with thecomplex electric field vector E at x ( t ), which describe thepolarization state, are defined as I = | E x | + | E y | (2) Q = | E x | − | E y | (3) U = 2Re( E x E ∗ y ) (4) V = 2Im( E x E ∗ y ) , (5)where I is the polarised fraction of the total intensity, L = ( Q + U ) / is the linearly polarised component, and V is the circularly polarised component. The observed elec-tric field vector E , assumed to be in the direction of theparticle acceleration , is the incoherent sum of the elec-tric field vectors at P and P , viz. E = E + E , wherethe relative phase between E and E fluctuates randomly.The observed Stokes parameters therefore reduce to I = I + I , Q = Q + Q , and U = U + U . In this paper, weassume that all the emission is linearly polarised for simplic-ity, i.e. V = 0. Circular polarization will be examined in acompanion paper.The x - and y - components are measured with respectto an orthonormal basis ( ˆx , ˆy ) which is fixed in the planeof the sky. In this paper, we choose ˆ x = Ω p / | Ω p | and ˆy =ˆ x × w , where Ω p = Ω − ( Ω · w ) w is the projection of Ω onto the sky. Then the polarization angle, ψ , between ˆ x and The instantaneous acceleration is inclined slightly with respectto the normal (or binormal) of B at x ( t ), because the emit-ting charges corotate. For more details, see the discussion aroundequation (2) in CM10 and equation (A3) in Dyks (2008).c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars the linearly polarised part of E is given by ψ = 12 tan − UQ . (6)The observational data obtained from the EPN are notnecessarily expressed in the canonical basis ( ˆx , ˆy ). However,the Q - U phase portrait has the same shape in any Carte-sian basis; if the basis is rotated by an angle β with respectto ˆx and ˆy , the Q - U phase portrait rotates by an angle2 β without being distorted, unlike the I - Q and I - U phaseportraits, which change shape. Hence, when analysing thedata, the first step is to reproduce the shape of the Q - U phase portrait as closely as possible without worrying aboutthe orientation; infer β ; rotate the ( I, Q, U, V ) data into thecanonical basis provisionally defined through β ; and then ad-just α, i , and the beam and polarization patterns iteratively to reproduce the I - Q and I - U portraits. The recipe for doingso is explained in Section 2.6 and Figure 3 of CM10. Figures 18–33 in the Appendix display look-up tables ofStokes phase portraits and PA swings, similar to those inCM10, updated to include interpulse emission. The figuresare organised into four groups, corresponding to two beammodels (filled core and hollow cone) and two polarizationmodels ( L ∝ cos θ , L ∝ sin θ ; see CM10). All the look-up ta-bles are for a current-modified dipole magnetic field (CM10)composed of a pure dipole plus a toroidal component withmagnitude B φ = − B p cos α sin θr/r LC , (7)where B p = ( B r + B θ ) / is the poloidal field strength. An interpulse is a secondary pulse separated from themain pulse by approximately 180 ◦ of rotational phase(Manchester & Lyne 1977). It is believed to arise when apulsar is a nearly orthogonal rotator viewed nearly side-on,shining from both magnetic poles, i.e. with α ≈ i ≈ ◦ ,where ‘ ≈ ’ means ‘within roughly one beam width’ in thiscontext (Petrova 2008).Figure 1 compares the Stokes phase portraits, pulse pro-files, and PA swings for a pure dipole and current-modifieddipole emitting from one and two poles for one illustrativeorientation ( α, i ) = (80 ◦ , ◦ ). For clarity, relativistic aber-ration is not included in this example (compare Section 2.4 et seq. ). Clockwise from the top left panel, the figure displays(i) a pure dipole with no interpulse, (ii) a current-modifieddipole at r = 0 . r LC with no interpulse, (iii) a pure dipolewith an interpulse, and (iv) a current-modified dipole at r = 0 . r LC with an interpulse. The top two panels in Fig-ure 1, which have no interpulse, are the same as in Figures5–8 and 30–33 in the look-up tables in CM10.Figure 2 shows the loci ˆ x ( t ) traced out by P and P over one rotation in the body frame of the pulsar for thevarious cases in Figure 1. The panels are arranged as in Fig-ure 1. The bottom panels, in which the interpulse is present,show two paths, one in the north hemisphere, and one in thesouth. For the current-modified dipole (right panels), the Figure 1.
Stokes tomography of a model pulsar with an inter-pulse but without relativistic aberration. Top left: dipole field,no interpulse. Top right: current-modified dipole emitting at r =0 . r LC , no interpulse. Bottom left: dipole field with interpulse.Bottom right: current-modified dipole emitting at r = 0 . r LC with interpulse. Within each quadrant of the figure, the five sub-panels display (clockwise from top left): I/I max and PA (in ra-dians) as functions of pulse longitude, I - Q , Q - U , and I - U . Theorientation is ( α, i ) = (80 ◦ , ◦ ). The beam pattern is given by(8). loci are asymmetric, as discussed in CM10. For definiteness,we consider a filled-core beam, viz. I ( θ, φ ) = (2 πσ ) − / (cid:8) exp (cid:2) − θ / (2 σ ) (cid:3) +exp (cid:2) − ( θ − π ) / (2 σ ) (cid:3)(cid:9) , (8)which is represented by greyscale shading in Figure 2. In (8), σ is the width of the beam, chosen arbitrarily to equal 10 ◦ .We also choose the linear polarization pattern to be L ( θ, φ ) = I ( θ, φ ) | cos θ | (9)in Figures 1 and 2. Other choices (e.g. L ∝ sin θ ) are equallyvalid and have been found empirically by CM10 to matchthe observational data in many objects.In Figure 1, the interpulse traces out a small, secondaryloop within the primary pattern in the I - Q , I - U and Q - U phase portraits. It also changes slightly the range of U and Q covered by the main pulse. For example, for the dipole(left panels), the maximum value of U decreases from 0.9to 0.8 with the addition of the interpulse. The size of thesecondary loop (i.e. the intensity of the interpulse) increasesrelative to the main pulse as α approaches i , as expected.Along α = 90 ◦ and i = 90 ◦ , when the interpulse and themain pulse peak at the same intensity, the primary and sec-ondary patterns overlap in the I - Q , I - U and Q - U planes,and the phase portraits are indistinguishable from the non-interpulse case. The shapes do not overlap exactly for otherorientations, where the main pulse is brighter than the in-terpulse.In the I - U and Q - U planes, the balloons and heart c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 2.
Locus ˆ x ( t ) traced out by the emission point(s) P and P over one rotation, in the body frame of the model pulsar con-sidered in Figure 1. Top left: dipole field, no interpulse. Top right:current-modified dipole emitting at r = 0 . r LC , no interpulse.Bottom left: dipole field with interpulse. Bottom right: current-modified dipole emitting at r = 0 . r LC with interpulse. Thebeam pattern I ( θ, φ ) is represented by greyscale shading (bright-ness ∝ I ). shapes seen in CM10 are also seen when an interpulse ispresent. For example, for α = i = 90 ◦ , the Stokes param-eters trace out two reflection-symmetric patterns with pos-itive and negative U to form complex, interlocking shapes(see Figures 18–32 from the atlas of look-up tables in theAppendix). As expected, the patterns are more intricate fora hollow cone than for a filled core. For example, the Q - U portrait at ( α, i ) = (70 ◦ , ◦ ) for a filled core contains anasymmetric, tilted heart shape and a small oval, both con-nected at Q = U = 0 (Figure 20). The same orientation fora hollow cone shows a broader heart shape with two large,secondary ovals (Figure 28). In the observer’s reference frame, charged particles flowingoutwards ultra-relativistically along poloidal magnetic fieldlines also have a small transverse velocity component be-cause they corotate with the star as part of the highly con-ducting magnetosphere. This displaces x ( t ) by a distanceof order r/r LC compared to its position when aberrationis neglected. The electric field vector (parallel to the parti-cle’s acceleration vector) is also displaced, resulting in thewell-known delay-radius relation (Blaskiewicz et al. 1991;Hibschman & Arons 2001; Dyks 2008). According to thisrelation, the centre of the pulse profile leads the steepestpoint of the PA swing by 4 r/r LC .We compute x ( t ) directly, including aberration, bysolving (1) numerically. As a cross check, we compare thenumerical solution with the analytic approximation givenby equation (F2) of Hibschman & Arons (2001), where thetangent field at the aberration-shifted emission point, t , canbe expressed as the sum of the tangent field at the origi- nal, non-aberrated emission point, t , plus a perturbation t = ( Ω × x ) /c − [( Ω × x ) /c · t ] t . As the aberration-induced deflection angle grows linearly with r , equation(F2) holds most accurately for small r . At r = 0 . r LC and 0 . r LC , the direct and approximate calculations of t agree to within ∼
10% and ∼
20% respectively. Note that,although we calculate t directly from (1), the expression ± t + Ω × x /c in (1) itself breaks down near the light cylin-der, where quadratic relativistic corrections come into play.Figure 3 illustrates how aberration modifies the Stokesphase portraits, pulse profiles, and PA swings for pure dipoleand current-modified dipole magnetospheres. For the sakeof clarity, we do not include interpulse emission in Figure3, although, in general, interpulse and aberration effects areadditive, as one can tell from the look-up tables in the Ap-pendix. The emission is placed arbitrarily at an altitude of0 . r LC to ensure a reasonably strong effect. We note thataberration introduces an altitude dependence in the case ofa pure dipole, which is absent in the non-aberrated dipoleconsidered in CM10.Aberration acts mainly to shift the relative phases ofthe pulse centroid and PA swing inflection point. To lowestorder in r/r LC , the radius-delay relation predicts that thepulse profile is phase shifted by ≈ − r/r LC radians, whereasthe PA swing is phase shifted by ≈ r/r LC radians. Figure3 shows that, for r = 0 . r LC , the pulse profile is shiftedby ≈ − ≈ +0.27 radians. These shifts are enough to dramaticallybroaden the I - Q pattern, twist the I - U pattern, and tilt the Q - U pattern for the pure dipole (left panels of Figure 3). Forthe current-modified dipole (right panels), the I - Q patternnarrows, the I - U patterns twists and rotates, and the Q - U pattern rotates. Figure 4 shows the loci of ˆ x ( t ) tracedout by P over one rotation (there is only one set of emissionpoints without an interpulse), with each panel correspondingto the cases in Figure 3. The loci of the aberrated emissionpoints (bottom panels) are shifted in φ relative to the non-aberrated points (top panels).In CM10, it is shown that, for a pure dipole field with-out aberration, all phase portraits are reflection symmetricabout U = 0. Aberration breaks this symmetry, causing theshapes in the I - U and Q - U plane to tilt (see Section 4 inCM10). Aberration also changes the tilt and the relativesizes of the shapes in the phase portraits, e.g. the ventriclesof the hearts in the U - Q plane. If the beam pattern is centred on another axis that is slightlytilted with respect to the magnetic axis (e.g. fan beams inthe outer magnetosphere; Cheng et al. 2000; Watters et al.2009), the pulse profile is also phase-shifted relative to thePA swing. Tilting the beam axis away from the magneticaxis can therefore mimic closely (though not exactly) theeffects of aberration (see Section 2.4.1 in CM10). To illus-trate, Figure 5 compares the phase-shift caused by the tiltof the beam axis to that caused by aberration. We plot sixpulse profiles and PA swings for a pure dipole with ( α, i ) =(30 ◦ , ◦ ). The three panels on the left correspond to beamaxes which are tilted with respect to the magnetic axis by( θ ′ , φ ′ ) = (10 ◦ , ◦ ) (top), (10 ◦ , ◦ ) (middle), and (10 ◦ , ◦ )(bottom), all emitting at r = 0 . r LC . Using the top panel c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 3.
Stokes tomography for a model pulsar including rel-ativistic aberration but without an interpulse. Top left: dipolefield, no aberration. Top right: current-modified dipole emittingat r = 0 . r LC , no aberration. Bottom left: dipole field emittingat r = 0 . r LC with aberration. Bottom right: current-modifieddipole emitting at r = 0 . r LC with aberration. Within each quad-rant of the figure, the five subpanels displays (clockwise from topleft): I/I max and PA swing (in radians) as a function of pulselongitude, I - Q , Q - U , I - U . The orientation is ( α, i ) = (30 ◦ , ◦ ).The beam pattern is given by (8). Figure 4.
Locus ˆ x ( t ) traced out by the emission point P acrossone rotation, in the body frame of the model pulsar considered inFigure 3. Top left: dipole field, no aberration. Top right: current-modified dipole emitting at r = 0 . r LC , no aberration. Bottomleft: dipole field emitting at r = 0 . r LC with aberration. Bot-tom right: current-modified dipole emitting at r = 0 . r LC withaberration. The beam pattern I ( θ, φ ) is represented by greyscaleshading (brightness ∝ I ). Figure 5.
Pulse profiles and PA swings for pure dipoles withidentical orientations ( α, i ) = (30 ◦ , ◦ ). The left panels showbeam axes offset by ( θ ′ , φ ′ ) with respect to the magnetic axis,corresponding to ( θ ′ , φ ′ ) = (10 ◦ , ◦ ) (top), ( θ ′ , φ ′ ) = (10 ◦ , ◦ )(middle), and ( θ ′ , φ ′ ) = (10 ◦ , ◦ ) (bottom). In the left panels,the emission radius is fixed at r = 0 . r LC . The right panels cor-respond to emission heights r = 0 . r LC (top), 0 . r LC (middle),and 0 . r LC (bottom). In the right panels, the magnetic and beamaxes are aligned. as a reference point, the pulse centroid leads the PA swingby − .
18 rad in the middle panel, and by − .
36 rad in thebottom panel. On the right-hand side of Figure 5, the beamaxis and magnetic axis are aligned, but we vary the emissionradius. As aberration causes the pulse centroid to lead thePA swing by − r/r LC , one could just as well attribute thephase shifts seen in the left-hand side to the emission radiusincreasing from r = 0 . r LC (top), 0 . r LC (middle), and0 . r LC (bottom).As it is not possible to distinguish the two effects with-out additional a priori information, throughout this paperwe assume the beam is centred on the magnetic axis, exceptin Sections 4 and 5, where we work with tilted beams re-constructed empirically from the pulse shape (because themagnetic-pole-centred model does not fit the data). We now survey the pulse profiles, Stokes phase portraits, andPA swings for a selection of 16 MSPs from the EPN onlinedatabase (Lorimer et al. 1998). These objects are chosenbecause they have pulse periods <
10 ms, except for PSRJ1022+1001, which was included because of its interestingStokes phase portraits and because its orientation angles α and i have been measured with some degree of confidence (cid:13) , 1–26 C. T. Y. Chung et al. by previous authors (Stairs et al. 1999). All objects were ob-served by Stairs et al. (1999), except for PSR J0437 − ψ defined in this paper corresponds to − ψ in Stairs et al.(1999). Table 3 quotes the size of the magnetosphere foreach MSP (in units of the stellar radius, r ⋆ ) and the frequen-cies where EPN data are available. In the few cases whererotating vector model fits have been attempted in the liter-ature, α and i are also quoted, together with the publisheduncertainties. In Figures 6–8, we present the Stokes phase portraits, pulseprofiles, and PA swings for the eight pulsars with the clean-est data. All Stokes parameters are normalized by the peakintensity I max . MSPs generally have a lower degree of lin-ear polarization than non-recycled pulsars, so their phaseportraits are correspondingly noisier. The PA swing is onlydrawn at pulse longitudes satisfying L > . L max and I > . I max , where L max is the peak value of L . As theabsolute orientation of Ω p (and hence the angle β betweenthe measured and canonical bases) for each set of data isunknown, we start by analysing just the shape of the Q - U phase portraits, as discussed in Section 2.1 and CM10.Many of the pulse and linear polarization profiles arehighly asymmetric, suggesting a complex emission pattern.In objects where I ( t ) and L ( t ) have multiple peaks or inter-pulse emission, the Stokes phase portraits feature multipleloops, each corresponding to an individual peak. For ex-ample, each of the five peaks seen in PSR J0437 − I - Q and I - U planes (see Section 5). In the Q - U plane, the pattern formed is an asymmetric figure-eightover a slightly curved line. PSR J1022+1001 (Figure 6, rows4–6) has an asymmetric, double-peaked pulse profile, whichproduces an asymmetric heart shape in the Q - U plane. Alsointeresting is PSR J1939+2134 (Figure 8, rows 4–5), whichhas a strong interpulse, whose phase portraits narrow withincreasing frequency, while those of the main pulse broaden.The PA swings for the MSPs featured in Figures 6–8 areless informative. In several cases, where the PA swing is flator noisy, the phase portraits still trace out a recognisablepattern. For example, in Figure 6, the PA swing of PSRJ0437 − − − The data in Figures 6–8 are too low in quality to allow de-tailed fits for the angles α and i and the magnetic geometry,except for PSR J1939+2134 and PSR J0437 − α ≈ i & ◦ . (ii) If there is more than one peak, wemodel the emission as a hollow cone. (iii) If L peaks with I ,we assume L = I cos θ , whereas, if L vanishes at the pulsecentroid, we assume L = I sin θ . Away from the α = i di-agonal, both linear polarization patterns yield similar phaseportraits.(i) PSR J1012+5307 (0.61 GHz, Figure 6, third row): hol-low cone, L = I cos θ , ( α, i ) = (70 ◦ , ◦ ) (Figures 26–29).This object has an interpulse. The balloons in the I - Q , I - U ,and Q - U planes match approximately the orientations of theballoons in the model, although they have different widths.(ii) PSR J1713+0747 (1.414 GHz, Figure 7, third row):filled core beam, L = I sin θ , ( α, i ) = (80 ◦ , ◦ ) (Figures18–21). The straight lines in all three phase portraits matchthe model, although the gradient of the I - Q line is less steepthan in the model.(iii) J1744 − L = I cos θ , ( α, i ) = (30 ◦ , ◦ ) (Figures 18–21).The straight line in the I - Q plane matches a thin balloonin the model, while the balloons in the I - U and Q - U planesmatch balloons in the model. Note that the balloons in themodel are tilted upwards ( dU/dQ > dU/dQ < − L = I cos θ , ( α, i ) = (60 ◦ , ◦ ) (Figures 26–29).The pulse profile has three peaks, suggesting that this objectmay have a double-peaked interpulse. The data matches themodel if the Q - U pattern is rotated by ≈ ◦ .Where multi-frequency observations are available, weonly analyse the frequency at which the phase portraits areresolved best. We emphasize that the matches are approx-imate, and that the figures in the Appendix show only thephase portraits at one altitude, viz. r = 0 . r LC . More de-tailed modelling of I and L as a function of emission altitudeand ( θ, φ ) must be done to obtain more accurate matches,including the possibility that the emission originates fromseveral altitudes (Johnston et al. 2008).For PSR J1022+1001 (Figure 6, rows 4–6), whose S-shaped PA swing is nominally dipolar, the heart shape inthe Q - U plane roughly matches a pure dipole at ( α, i ) ≈ (70 ◦ , ◦ ) for a hollow cone with either polarization model.However, the observed heart shape differs slightly fromthe model, and L ( t ) is actually triple-peaked, not double-peaked. Further information on the polarization basis (e.g.at several frequencies) is required in order to accurately de-termine the orientation and magnetic geometry.We now test whether the published α and i values inTable 3, inferred from the rotating vector model, are con- c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Table 1.
Pulse periods P and observation frequencies of 15 millisecond pulsars from the EPN online database with P <
10 ms (except forJ1022+1001, which is included because of its interesting Stokes phase portraits). Where a rotating vector model fit has been publishedpreviously, α and i are quoted with their published uncertainties. The uncertainties for α and i for PSR J0437 − − P (ms) r LC /r ⋆ Frequency (GHz) α ( ◦ ) i ( ◦ )J0034 − ±
11 —0.61 8 ±
15 —J0437 − − ±
16 135 . ± . ±
27 75 . ± . − − − − − . ± . . ± . − − sistent with the observed Stokes phase portraits. We referthe reader to the look-up tables in Figures 18–32 in the Ap-pendix. For PSR J1022+1001 (Figure 6, rows 5–6), the PAswings at 0.61 GHz and 1.414 GHz imply two very differentorientations, namely, ( α, i ) = (140 ◦ , ◦ ) and (83 ◦ , ◦ ) re-spectively (Stairs et al. 1999). Already, this is worrying, asthe orientation of a given pulsar should be unique, no mat-ter what altitude the emission comes from. Moreover, nei-ther of these orientations yield Stokes phase portraits whichmatch the data, for any beam or linear polarization pattern.One can verify this easily by examining the phase portraitsin the vicinity of ( α, i ) = (40 ◦ , ◦ ) and (80 ◦ , ◦ ) in Fig-ures 18–32 in the Appendix. For example, for a hollow conewith L = I sin θ , at (80 ◦ , ◦ ), there are three interlock-ing ovals in Q - U , unlike the heart shape in Figure 6. ForPSR J1824 − α, i ) = (41 ◦ , ◦ ). For a hollow cone, thelook-up tables in Figure 26–28 show interlocking ovals inthe Q - U plane, whereas the data reveal an oval joined toa straight line. Significantly, all these discrepancies are inthe shape, not the orientation of the Q - U portrait, which isbasis-independent (CM10). We comment on the published( α, i ) fits for J0437 − We now discuss how the pulse profiles and Stokes phaseportraits evolve with frequency for PSR J1022+1001 and PSR J1939+2134, the only EPN MSPs with adequate multi-frequency data.At 0.41 GHz, PSR J1022+1001 has a double-peakedintensity profile. The first intensity peak is itself double-peaked in terms of its linear polarization, resulting in three L peaks overall. In the phase portraits, the first intensitypeak corresponds to the bottom loop of the figure-eight inthe I - Q plane and the large balloon in the I - U plane. At0.61 GHz, the second pulse is stronger than the first. In the I - U plane, the second pulse corresponds to the long, straighttail emerging from the bottom of the balloon.At 0.61 GHz, PSR J1939+2134 displays a single-peakedinterpulse which peaks at ≈ . I max . The main pulse is alsosingle peaked. The two pulses trace out qualitatively similarpatterns on the I - Q and I - U planes, namely elongated bal-loons, whose major axes are tilted by ≈ ◦ and ≈ ◦ relativeto the Q = 0 and U = 0 axes respectively. At 1.414 GHz,the peak of the interpulse drops to ≈ . I max , and the mainpulse is double-peaked. There is a dramatic difference in Q for the main pulse: the slope of the major axis of the bal-loon changes sign, from dQ/dI < dQ/dI > I - Q and I - U balloons.For all the MSPs, the different ways in which individ-ual peaks evolve with frequency imply that I and L dependon θ and φ in a complicated way. The profile componentsmight originate from different emission regions whose mag-netic geometries are different functions of r . As the aberra- c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 6.
Pulse profiles, Stokes phase portraits and PA swings for PSR J0437 − I/I max (solid curve) and
L/I max (dashed curve)versus time (in s), (2) the I - Q phase portrait, (3) the I - U phase portrait, (4) the Q - U phase portrait, (5) the PA swing versus time (ins) (data points with L > . L max and I > . I max plotted only). Data are presented courtesy of the EPN online archive.c (cid:13)000
L/I max (dashed curve)versus time (in s), (2) the I - Q phase portrait, (3) the I - U phase portrait, (4) the Q - U phase portrait, (5) the PA swing versus time (ins) (data points with L > . L max and I > . I max plotted only). Data are presented courtesy of the EPN online archive.c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 7.
Pulse profiles, Stokes phase portraits and PA swings for PSR J1713+0747 at 0.41 GHz, 0.61 GHz and 1.414 GHz, PSRJ1744 − − − I/I max (solid curve) and
L/I max (dashed curve) versus time(in s), (2) the I - Q phase portrait, (3) the I - U phase portrait, (4) the Q - U phase portrait, (5) the PA swing versus time (in s) (datapoints with L > . L max and I > . I max plotted only). Data are presented courtesy of the EPN online archive.c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 8.
Pulse profiles, Stokes phase portraits and PA swings for PSR J1911 − I/I max (solid curve) and
L/I max (dashed curve) versus time (in s), (2) the I - Q phase portrait, (3) the I - U phase portrait, (4) the Q - U phase portrait, (5) the PA swing versus time (in s) (data points with L > . L max and I > . I max plotted only). Data are presentedcourtesy of the EPN online archive. c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars tion and toroidal field increase with r , they also distort thepath ˆ x ( t ), further complicating I ( t ) and L ( t ). In this section and the next, we model the pulse andlinear polarization profiles of PSR J1939+2134 and PSRJ0437 − P = 1 .
558 ms and ˙ P =1 . × − s s − (Kaspi et al. 1994), making it the second-fastest known MSP. Data for this object, at 0.61 GHz and1.414 GHz, are obtained from the EPN online archive. Thedata were originally published in Stairs et al. (1999). Wechoose this object because of its strong interpulse emission.As mentioned in CM10, the data published in the EPN arenot expressed in the canonical polarization basis describedin Section 2. Additionally, the emission altitude for this ob-ject has not been estimated at either frequency due to theflatness of its PA swing. We are therefore obliged to infer β indirectly, from qualitative considerations, in order to bringthe data of both frequencies into the canonical basis.To accomplish this, we make a few general observa-tions, which provide insight into the magnetic geometryand emission pattern. Firstly, we note that the shape of themain pulse changes significantly from an asymmetric single-peaked profile at 0.61 GHz to an asymmetric double-peakedprofile at 1.414 GHz. This suggests that the magnetic geom-etry and possibly the beam pattern change with emissionaltitude. Both pulse profiles, however, are consistent withhollow cone emission. Secondly, in the main pulse, the lin-ear polarization follows the total intensity closely, suggestingthat L ∝ cos θ is a reasonable approximation. This is alsotrue to a lesser degree in the interpulse.The stellar surface of this object lies at 0 . r LC , placinga lower limit on the emission altitude. If we assume arbitrar-ily that the data at 0.61 GHz are already in the canonicalbasis, we find that, at ( α, i ) = (20 ◦ , ◦ ), the models for bothpure and current-modified dipoles exhibit tilted balloons inthe I - U and Q - U planes, similar to the data at r ≈ . r LC .Unfortunately, without additional information on the abso-lute orientation of Ω p , we are limited to this assumption.Figures 9 and 10 show the pulse profile, PA swing, andStokes phase portraits of (a) the main pulse and (b) theinterpulse at 0.610 GHz and 1.414 GHz respectively. For the0.610 GHz case, we assume β = 0, whereas for the 1.414 GHzcase, we align the narrow balloon shape in the Q - U plane ofFigure 10(a) with that of Figure 9(a) by assuming β = 45 ◦ .In the top left panel of each subfigure, we plot I/I max (solidcurve),
L/I max (dashed curve) and the PA swing (dottedcurve) wherever L > . L max . Stepping clockwise, the nextthree panels show I - Q , Q - U and I - U .We now examine the magnetic geometry, beam pattern,orientation, and emission altitude in more depth in Sections4.1–4.4. We find that the model with a hollow cone and L ∝ cos θ must be generalized by letting I and L vary with φ in order to fit the data in detail. At 0.61 GHz, the Stokes phase portraits for the main pulseare all narrow balloons. The major axes of the balloons tiltin different directions: we find dQ/dI < I - Q plane, dU/dI < I - U plane, and dU/dQ > Q - U plane. Assuming β = 0, the tilt of the Q - U phase portraitdiscounts a pure dipole magnetosphere at a low emissionaltitude. At higher altitudes, where aberration is important,e.g. at r = 0 . r LC , the pure and current-modified dipolesproduce phase portraits that are similar enough at someorientations to warrant considering both cases.We now seek a match from the look-up tables for ahollow cone with L = I cos θ (Figures 26–28), keeping inmind that we are interested in orientations which provide aninterpulse ( α or i & ◦ ). We find that, at ( α, i ) = (20 ◦ , ◦ ),the phase portraits for I - Q and I - U match approximatelythe balloons in the data, although the sign of U is reversed(this choice of orientation is justified in Section 4.2). ( α, i )Finding the exact orientation is an iterative process, requir-ing the beam and polarization patterns to be adjusted ateach step. Initially, we seek a match to the data at 0.61 GHz(Figure 9). The interpulse is extremely useful in narrowingthe range of possible orientations to 70 ◦ . α, i . ◦ (as-suming beams of intrinsically equal luminosity). As the in-terpulse is weaker than the main pulse, we know that α and i are less than 90 ◦ . From Figures 26–28, there are two ori-entations with similar balloons in all three phase portraits,namely ( α, i ) = (20 ◦ , ◦ ). The phase portraits for a puredipole at ( α, i ) = (20 ◦ , ◦ ) are also similar.Before zooming in to refine the grid around ( α, i ) =(20 ◦ , ◦ ), we experiment with various emission altitudeswhile tailoring the pattern to fit the data. In Section 4.3,we construct beam and linear polarization patterns at r =0 . r LC for the data at 0.61 GHz. Both the main pulse and interpulse at 0.61 GHz are single-peaked and skewed to the left. The interpulse peaks at ≈ . I max . To capture this behaviour, we model the emissionregion as two hollow cones whose brightness varies longitu-dinally, i.e. the cones are shaped like horseshoes in cross-section. The best-fit beam pattern is given empirically by I ( θ, φ ) = (2 πσ ) − / [0 . | sin( φ − . | ] × exp (cid:2) − . θ − ρ ) /σ (cid:3) (10)+0 . πσ ) − / [0 . | sin( φ − . | ] × exp (cid:2) − . θ − π + ρ ) /σ (cid:3) , (11)where σ = 3 ◦ and σ = 3 . ◦ are the widths of the mainpulse and interpulse respectively, and ρ = 23 ◦ and ρ = 35 ◦ are the corresponding opening angles. The modelled pulsesare ≈ c (cid:13) , 1–26 C. T. Y. Chung et al. (a) Main pulse (b) Interpulse
Figure 9.
Polarimetry of (a) the main pulse and (b) the interpulse of PSR J1939+2134 at 0.61 GHz (Stairs et al. 1999). Each subfigureshows (clockwise from top left panel): (i)
I/I max (lower subpanel, solid curve) and
L/I max (lower subpanel, dashed curve) profiles andPA swing (upper subpanel, dotted curve, in rad) versus pulse phase (in degrees); (ii) I - Q phase portrait; (iii) Q - U phase portrait; (iv) I - U phase portrait. Data are presented courtesy of the EPN.(a) Main pulse (b) Interpulse Figure 10.
As for Figure 9 but at 1.414 GHz (Stairs et al. 1999). Data are presented courtesy of the EPN. L follows the pulse profile closely, lagging the pulse centroid inphase by ≈ . ◦ , peaking at ≈ . I max . In the interpulse, L is extremely low, peaking at ≈ . I max , and appears to betriple-peaked. Despite the apparent difference in the profiles,we are able to reproduce them surprisingly well using thesame model, given by L ( θ, φ ) = | cos θ sin( φ + 0 . | (12)without invoking a north-south asymmetry. As expected, however, (12) reproduces the L profile of the main pulsemore accurately than that of the interpulse. We emphasizethat (11) and (12) are certainly not unique and do not fitthe data exactly, but they are adequate for the empiricaltask at hand.Adopting (11) and (12), we generate zoomed-in look-up tables for both pure and current-modified dipoles, in therange 14 ◦ α ◦ , 76 ◦ i ◦ , with a resolution of 2 ◦ .We find the closest match is for a current-modified dipoleat ( α, i ) = (22 ◦ , ◦ ), with a ‘by eye’ uncertainty of ± ◦ for α and ± ◦ for i . This margin would widen if I ( θ, φ ) and L ( θ, φ ) were adjusted for each orientation.In Figure 11, we plot the pulse profile, PA swing andStokes phase portraits of the model at r = 0 . r LC and c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars ( α, i ) = (22 ◦ , ◦ ). The slight jaggedness of the pulse pro-files is a product of the finite grid resolution. The Stokesphase portraits of the main pulse [Figure 11(a)] match thedata in Figure 9(a) reasonably well. In the data, the I - Q balloon ranges from − . . Q .
0, whereas in the modelit is thinner and ranges from − . . Q .
0. The I - U bal-loon in the data ranges from − . . U .
0, whereas in themodel it ranges from − . . U .
0. The PA swing in thedata is nearly flat, with a slight negative gradient, whereasthe model shows a slight positive gradient. For the inter-pulse, there is poorer agreement in L . The tilted balloonin I - Q from the data [Figure 9(b)] is reproduced in Figure11(b), including the kink visible at ( I, Q ) ≈ (0 . , − . I - U plane, the data feature a tilted balloon, with aprominent kink at ( I, U ) ≈ (0 . , − . U = 0, it would match more closely. In the Q - U plane, thetilted oval seen in the data is reproduced in the model, butwith − . . U . .
05 in the data, and − . . U . According to the standard radius-to-frequency mapping, theobservation frequency scales with emission radius as r − / (Ruderman & Sutherland 1975; Cordes 1978). If the data at0.61 GHz correspond to r = 0 . r LC , then 1.414 GHz corre-sponds to r = 0 . r LC . Figures 12–13 show the pulse pro-files, PA swing and Stokes phase portraits predicted theo-retically for both pulses, for emission altitudes ranging from r = 0 . r LC to 0 . r LC . The relative heights of the pulseschange with emission altitude. We label them Pulse 1 (Fig-ure 12), corresponding to the main pulse in the data, andPulse 2 (Figure 13), corresponding to the interpulse in thedata.The theoretical pulse profile and phase portraits at r = 0 . r LC (Figures 12–13, top row) display some inter-esting features. First, the main pulse and interpulse haveroughly the correct shapes, but swap positions in phase, i.e.the hollow cone which emits the main pulse at r = 0 . r LC also emits the interpulse at r = 0 . r LC , and vice versa.Upon inspection, it is likely that the same is true in thedata. The triple-peaked linear polarization profile seen inFigure 9(b) is also present in Figure 10(a), although thefirst component in L is much weaker than the second andthird at 1.414 GHz. Additionally, the kinks seen in the I - Q and I - U planes of Figure 9(b) are seen in Figure 10(a). Inthe data, the interpulse peaks at ≈ . I max , compared to ≈ . I max in the model at r = 0 . r LC .Second, the linear polarization profile and phase por-traits at r = 0 . r LC reproduce the main pulse reasonablywell but do not match the interpulse. The linear polariza-tion of the main pulse (Pulse 1; Figure 12, top row) is ≈ L peak in the main pulse is compara-ble in height to the second peak, while in the data it isweaker. In the simulated I - Q and I - U planes of the mainpulse, there are reasonable matches to the balloons in the data. In the simulated I - Q plane, we see a balloon witha kink at ( I, Q ) ≈ (0 . , − . I, Q ) ≈ (0 . , − . I - U plane, the kinkseen in the data at ( I, U ) ≈ (0 . , − .
1) is reproduced at(
I, U ) ≈ (0 . , − .
2) in the model. In the Q - U plane, thedata trace out a thin balloon with dU/dQ > − . . Q . − . . U . .
05. The simulated phaseportrait shows a thin balloon with the same orientation,spanning − . . Q . − . . U .
0. For the in-terpulse (Pulse 2; Figure 13, top row) the simulated totalintensity is twice the observed intensity, and the simulatedlinear polarization is ≈ I - Q and I - U planes are rotated by 90 ◦ clockwise with respect tothe data, whereas the Q - U balloon is rotated by 180 ◦ . Thesediscrepancies are also reflected in the PA swing.As the emission altitude increases from r = 0 . r LC to r = 0 . r LC , the phase portraits of the main pulse (Pulse1; Figure 12) change. In the I - Q plane, the kink in the bal-loon shifts towards ( I, Q ) ≈ (0 . , − . r = 0 . r LC ,the balloon in the I - U plane starts to resemble the hockeystick seen in Figure 11(b). In Q - U , the thin balloon rotatesclockwise. In the interpulse (Pulse 2; Figure 13), the I - Q balloon narrows and lengthens in Q , and the Q - U balloonnarrows.From Figures 12–13, we draw the following conclusions.(i) Although the simple model given by (11) and (12) mod-els the 0.61 GHz data reasonably successfully, it fails for thedata at 1.414 GHz displayed in Figure 10. However, the ob-served pulse profiles and Stokes phase portraits suggest thatthe emission region of the main pulse at 0.61 GHz corre-sponds to that of the interpulse at 1.414 GHz, and vice versa.Additionally, the emission pattern may change with radius.(ii) The discrepancies between the data and the phase por-traits at r = 0 . r LC (the altitude predicted by the radius-to-frequency mapping) indicate that the toroidal field maynot increase monotonically with r . The phase portraits forthe interpulse between r = 0 . r LC and r = 0 . r LC are alla poor match to the data. (iii) It is possible that the datashould be referred to a different value of β at 1.414 GHzthan the one we assume, which would rotate the Q - U phaseportrait, and change the shapes of the I - Q and I - U patterns. − We now repeat the procedure in Section 4 and CM10 forPSR J0437 − P = 5 .
758 ms and ˙ P =5 . × − s s − (Bell et al. 1997). It was chosen becauseit exhibits five distinct peaks in its pulse profile, clearly vis-ible at 1.44 GHz, and a highly structured PA swing. Thereis no interpulse observed in this object. Unlike the otherobjects considered in this paper and CM10, we find thatPSR J0437 − I ( t ) data ex-actly. Indeed, the Stokes phase portraits point persuasivelyto the existence of a strong quadrupole and higher-ordermultipoles at the radio emission altitude. In this respect,PSR J0437 − c (cid:13) , 1–26 C. T. Y. Chung et al. (a) Main pulse (b) Interpulse
Figure 11.
Theoretical polarization model of (a) the main pulse and (b) the interpulse of PSR J1939+2134 for a current-modified dipoleemitting at r = 0 . r LC , with orientation ( α, i ) = (22 ◦ , ◦ ), beam pattern given by (11), and linear polarization given by (12). Eachsubfigure shows (clockwise from top left panel): (i) I/I max (lower subpanel, solid curve) and
L/I max (lower subpanel, dashed curve)profiles and PA swing (upper subpanel, dotted curve, in rad) versus pulse phase (in degrees); (ii) I - Q phase portrait; (iii) Q - U phaseportrait; (iv) I - U phase portrait. restrict ourselves to presenting the argument that the pureand current-modified dipoles categorically fail to match thedata for the polarization models that work well for the otherobjects studied in this paper and CM10.Data for PSR J0437 − I/I max (solid curve),
L/I max (dashed curve) and the PA swing (dot-ted curve, at longitudes where L > . L max ). Steppingclockwise, the next three panels show I - Q , I - U and Q - U .As with PSR 1939+2134, we assume that the data ob-tained from the EPN are presented in the canonical polar-ization basis at one reference frequency, chosen here to be1.44 GHz. The stellar surface is at r = 0 . r LC , providinga lower limit on the emission altitude.The pulse profile has five components, labelled A–Ein Figure 14, two on either side of the largest peak (C).Each component in the pulse profile corresponds to a dis-tinct sub-pattern in the Stokes phase portraits. In the I - Q plane, peaks A and B correspond to the two small loops at Q >
0, peak C is the large figure-eight, and peaks D and Ecorrespond to the loop at
Q <
0. In the I - U plane, peaksA and B correspond to the kinks at U <
0, peak C is thelarge balloon, and peaks D and E correspond to the kinkat
U >
0. The U - Q plane is complicated, forming a roughX-shape, with one diagonal having dU/dQ > dU/dQ < Q >
Q <
Figure 14.
Polarimetry of PSR J0437 − I/I max (lower subpanel, solid curve) and
L/I max (lower sub-panel, dashed curve) profiles, and PA swing (upper subpanel, dot-ted curve, in rad) all plotted against pulse phase (in degrees); (b) I - Q phase portrait; (c) Q - U phase portrait; (d) I - U phase por-trait. Data are presented courtesy of the EPN. The linear polarization within peak C is double-peaked.This kind of structure is common and is modelled adequatelyby a filled core beam with L = I sin θ , as demonstrated forseveral objects in CM10. The phase separation of the peakssuggests that A, B, D, and E originate from two hollow conescentred on the same axis (peak C). Peak pairs B/C and C/D c (cid:13)000
L/I max (lower sub-panel, dashed curve) profiles, and PA swing (upper subpanel, dot-ted curve, in rad) all plotted against pulse phase (in degrees); (b) I - Q phase portrait; (c) Q - U phase portrait; (d) I - U phase por-trait. Data are presented courtesy of the EPN. The linear polarization within peak C is double-peaked.This kind of structure is common and is modelled adequatelyby a filled core beam with L = I sin θ , as demonstrated forseveral objects in CM10. The phase separation of the peakssuggests that A, B, D, and E originate from two hollow conescentred on the same axis (peak C). Peak pairs B/C and C/D c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 12.
Theoretical polarization model of Pulse 1 of PSR J1939+2134 as a function of emission altitude, for a current-modifieddipole with ( α, i ) = (22 ◦ , ◦ ). In landscape mode, the plots for each emission altitude occupy rows, increasing from r = 0 . r LC (toprow) to r = 0 . r LC (bottom row). From left to right, the columns show (1) I/I max (solid curve) and
L/I max (dashed curve) versuspulse phase l (in units of degrees), (2) I - Q phase portrait, (3) I - U phase portrait, (4) Q - U phase portrait, and (5) the PA swing (in rad;data points with L > . L max plotted only).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 13.
As for Figure 12 but for Pulse 2 of PSR J1939+2134. are separated by ≈ . ≈ . ≈ . α and i lie in ranges where interpulse emissiondoes not contribute significantly. In order to determine the magnetic geometry, we search thelook-up tables for a good match involving a filled core anda hollow cone beam. We assume L = I sin θ because of thedouble-peaked L profile in peak C. The phase portraits forthe filled core should match the large patterns correspondingto peak C, while the phase portraits for the hollow cone c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars should match the smaller patterns. We do not expect perfectmatches due to the complex beam and polarization patterns.At a minimum, however, we seek an approximate matchfor the rough figure-eight that forms diagonal 1 in the Q - U plane, the figure-eight in I - Q , and the balloon in I - U .The pure dipole look-up tables at r = 0 . r LC do notfeature figure-eight shapes in the I - Q plane for any orienta-tion. For α < i , there are balloons in I - U , and heart shapesin Q - U . As in many other opbjects, a pure dipole is ruledout (CM10).For the current-modified dipole (Figures 22–24), theclosest match is at ( α, i ) = (30 ◦ , ◦ ) (see Section 5.2 for adetailed justification). For a filled core, there are asymmetricfigure-eights in the Q - U and I - U planes (Figures 22–24), anda broad oval in I - Q (Figure 22). The hollow cone phase por-traits at this orientation feature asymmetric mosquitoes in I - Q and I - U (Figure 30–31) and an asymmetric heart shapein Q - U (Figure 32). At this stage, we cannot confidently dis-count either magnetic configuration, as the Stokes portraitschange when the filled core and hollow cones are combined.This issue is examined thoroughly in Sections 5.3–5.4. In this section, we justify ( α, i ) = (30 ◦ , ◦ ) as the bestmatching orientation. As PSR J0437 − α > ◦ , i > ◦ . We also rule out orientations with i > α because the associated phase portraits look nothing like thedata. For example, for a filled core, Figures 22–24 containballoons in I - Q , narrow, tilted balloons and straight lines in I - U , and ovals in U - Q . None of these patterns appear in thedata in Figure 14. The Q - U discrepancy is especially signif-icant as the shape of the Q - U portrait is basis-independent.The best matching orientation predicted by the RVMis ( α, i ) = (145 ◦ , ◦ ) (Manchester & Johnston 1995), al-though the authors note that the PA swing deviates largelyfrom the model. For this reason, no formal uncertainties areassigned to the fitted parameters, which were chosen to bea reasonable representation of the data. For a dipole field,the Stokes phase portraits are symmetric about ( α, i ) =(90 ◦ , ◦ ), i.e. the phase portraits for ( α, i ) = (145 ◦ , ◦ )and (35 ◦ , ◦ ) are identical. From CM10, the phase portraitsfor a pure dipole at ( α, i ) = (40 ◦ , ◦ ) and r ≪ r LC , witha hollow cone and L = sin θ , feature a narrow, tilted bal-loon in I - Q , a mosquito in I - U , and a heart in Q - U (seeFigures 22–24 in CM10). The I - U and Q - U shapes are sym-metric about U = 0. Interestingly, although we have chosenour best match independently of the RVM results, the twoorientations are close.We reiterate that the complex multiple-peaked beamand polarization patterns complicate the matching process.Some orientations must be tested with beam patterns tai-lored to fit the data, as described in Section 5.3, before beingruled out. For ( α, i ) = (30 ◦ , ◦ ), the appropriate beam pat-tern is a filled core surrounded by two hollow cones. Theresulting phase portraits show distorted, tilted balloons inboth I - Q and I - U . These shapes resemble roughly the datain Figure 14, although there are large discrepancies too,chiefly that the figure-eight in I - Q is missing, and that I - U is not symmetric about U = 0. In Sections 5.3 and 5.4, we construct detailed beam and linear polarization models inan attempt to improve the fits. In fitting the complex pulse profile of PSR J0437 − r LC –0.3 r LC .We consider a simpler model and focus on one fixedaltitude. We model the pulse profile empirically with a filledcore, I ( θ, φ ) (peak C), surrounded by two hollow cones, I ( θ, φ ) (peaks B and D) and I ( θ, φ ) (peaks A and E). Theintensity maps take the empirical forms I ( θ, φ ) = (2 πσ ) − / (cid:8) exp (cid:2) − θ / (2 σ ) (cid:3) +exp (cid:2) − ( θ − π ) / (2 σ ) (cid:3)(cid:9) , (13) I ( θ, φ ) = β ( φ )(2 πσ ) − / (cid:8) exp (cid:2) − ( θ − ρ ) / (2 σ ) (cid:3) +exp (cid:2) − ( θ − π + ρ ) / (2 σ ) (cid:3)(cid:9) , (14) I ( θ, φ ) = β ( φ )(2 πσ ) − / (cid:8) exp (cid:2) − ( θ − ρ ) / (2 σ ) (cid:3) +exp (cid:2) − ( θ − π + ρ ) / (2 σ ) (cid:3)(cid:9) , (15)where σ = 6 . ◦ , σ = 2 . ◦ and σ = 2 ◦ are the beam widthsof the core and cones, ρ = 18 ◦ and ρ = 26 ◦ are the openingangles of the two cones, and β ( φ ) and β ( φ ) are functionsdescribing the longitudinal structure of the two cones, givenempirically by β ( φ ) = 0 . | cos(0 . φ ) | , (16) β ( φ ) = 0 . | cos(0 . φ ) | . (17)As in Section 4, the cones are shaped like horseshoes. Given(13)–(15), we also find that the linear polarization patternis fitted empirically by L ( θ, φ ) = (3 θ ) − sin( θ − .
01) + 0 . | cos(0 . φ ) |× (2 πσ ) − / exp (cid:2) − ( θ − ρ ) / (2 σ ) (cid:3) . (18)We emphasize that equations (13)–(18) are not uniquefits, nor do they produce perfect agreement with the data.In particular, the data show that the B/C peaks are closerto each other than C/D, yet we are unable to reproducethis with a reasonably simple model. Our modelled peaksare equidistant. The models are sensitive to the pulsar’sorientation. Every time we vary α or i around (30 ◦ , ◦ ),we must adjust the coefficients in (13)–(18). We find thatthe closest match to the data, although poor, is achieved at( α, i ) = (32 ◦ , ◦ ). We note as well that the Stokes phaseportraits match marginally better if we rotate the polariza-tion basis by β = 90 ◦ . The difficulty in achieving a goodmatch may well be telling us that the underlying magneticgeometry is not a current-modified dipole. We now demonstrate how the Stokes phase portraits changeas we add I and I to the filled core I . Figure 15 shows thepulse profile, PA swing, and phase portraits for I , I + I ,and I + I + I respectively at an emission altitude of r =0 . r LC and β = 90 ◦ . c (cid:13) , 1–26 C. T. Y. Chung et al.
For just the filled core (Figure 15, top row in landscapeorientation), corresponding to peak C in the data, there isa thin, tilted balloon in I - Q , a balloon in I - U , and a tiltedoval in Q - U . Aside from the obvious dissimilarity with thefigure-eight seen in the data in I - Q , the oval in Q - U does notresemble diagonal 1 in the data. The balloon in I - U is anapproximate match to the data, although it is not symmetricabout U = 0.For the filled core and one hollow cone (Figure 15, mid-dle row), i.e. peaks B, C and D in the data, the conal com-ponents introduce a kink at ( I, Q ) ≈ (0 . , − . I - U plane, kinks are predicted to occur at I . .
2. The dataalso contain kinks in this region. In the Q - U plane, there isanother kink near ( Q, U ) ≈ (0 , − . I - Q and I - U inthe region I . .
15. In Q - U , a secondary oval forms. At astretch, it might be said that this secondary oval correspondsto one diagonal of an X-shape while the other large ovalcorresponds to another, but other interpretations are equallypossible. Adjusting the emission altitude does not improvethe fit.For completeness, in Figure 16, we present the pulseprofile, PA swing, and phase portraits for a filled core withtwo hollow cones for a pure dipole field at the same ori-entation and altitude, and β = 0. The phase portraits arealso a poor match to the data. The I - Q plane features atilted, asymmetric balloon, whereas the I - U plane featuresan asymmetric figure-eight. In the Q - U plane, there is a dis-torted oval surrounded by a tilted heart shape. Again, thebasis-independent Q - U shape does not resemble the data atall. The kinks seen in the phase portraits of the current-modified dipole also appear in the pure dipole.Finally, in Figure 17 we present phase portraits for asimplified version of the current-modified field where thetoroidal field is given by B φ = − B p r/r LC . (19)In this stripped-down expression, B φ depends on θ and α only through the poloidal field, and scales simply as r/r LC .The phase portraits in FIgure 17 are also presented for afilled core with two hollow cones at the same orientationand altitude, and β = 0. Again, they are a poor match tothe data, although the I - Q plane now features a tilted figure-eight similar to the data. The I - U plane features an asym-metric figure-eight, and the Q - U plane features two inter-locking ovals. As in the previous cases, kinks correspondingto the various pulse peaks punctuate the phase portraits.We note that, for the three magnetic geometries con-sidered, the theoretical PA swings are smooth and relativelyflat and do not contain any of the kinks seen in the data.Attempts to rotate the Q - U portraits of the pure and simpli-fied current-modified dipole to yield a better fit, i.e. triallingseveral values of β , are also unsuccessful. For example, for β = 150 ◦ with the simplified current-modified dipole, diag-onal 1 in the Q - U plane aligns with the large, interlockingovals in the model while diagonal 2 aligns with the smaller,third oval. However the I - Q plane now features a large bal- Figure 16.
Theoretical polarization model of PSR J0437 − r = 0 . r LC with ( α, i ) = (32 ◦ , ◦ ),beam pattern given by (13), (14) and (15), and linear polarizationgiven by (18). Clockwise from top left panel: (a) I/I max (lowersubpanel, solid curve) and L (lower subpanel, dashed curve) pro-files, and PA swing (upper subpanel, dotted curve, in rad) allplotted against pulse longitude (in units of degrees) (b) I - Q phaseportrait; (c) Q - U phase portrait; (d) I - U phase portrait. loon symmetric about Q = 0, and the I - U plane features abroad, tilted balloon, neither of which matches the data.We conclude that we are unable to fit the Stokes phaseportraits for PSR J0437 − α, i, I ( t ) and L ( t ) are correct. (iii) The magnetic field inthe emission region is neither a pure nor a current-modifieddipole (very likely). In this paper, we generalize the Stokes tomography tech-nique introduced by CM10 by adding interpulse emission. InSection 3, we present the Stokes phase portraits of 15 MSPsfrom the EPN online archive. By comparing the data to thegeneralized look-up tables for a current-modified dipole inthe Appendix, we are able to infer approximately the ge- c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 15.
Bottom-up, component-wise assembly of theoretical polarization models for PSR J0437 − r = 0 . r LC with ( α, i ) = (32 ◦ , ◦ ) and β = 90 ◦ . In landscape orientation, from top to bottom, we plot the following beampatterns: (top row) peak C, filled core, equation (13); (middle row) peaks B–D, filled core plus hollow cone, equations (13) and (14); and(bottom row) peaks A–E, filled core plus two hollow cones, equations (13), (14) and (15). Linear polarization in all three rows is givenby (18). From left to right, in landscape orientation, the columns contain (1) I/I max (solid curve) and
L/I max (dashed curve) profiles,plotted against pulse longitude (in units of degrees); (2) I - Q phase portrait; (3) I - U phase portrait; (4) Q - U phase portrait, and (5) PAswing (dotted curve, in rad).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 17.
Theoretical polarization model of PSR J0437 − r = 0 . r LC with ( α, i ) = (32 ◦ , ◦ ), beam pattern given by (13),(14) and (15), and linear polarization given by (18). Clockwisefrom top left panel: (a) I/I max (lower subpanel, solid curve) and L (lower subpanel, dashed curve) profiles, and PA swing (uppersubpanel, dotted curve, in rad) all plotted against pulse longi-tude (in units of degrees) (b) I - Q phase portrait; (c) Q - U phaseportrait; (d) I - U phase portrait. ometric orientations for five of the MSPs. This is an im-provement on the PA swing and rotating vector model,which yield orientations for only two of the objects —orientations which, it transpires, are inconsistent with theobserved Stokes phase portraits. In Section 4, we modelPSR J1939+2134 in detail, obtaining a match for the dataat 0.61 GHz with a current-modified dipole for ( α, i ) =(22 ± ◦ , ± ◦ ) and r = 0 . r LC . However, we are unable toreproduce the data at 1.414 GHz for the same orientation ataltitudes in the range 0 . r LC r . r LC . In Section 5,we repeat the process with PSR J0437 − α, i ) = (32 ◦ , ◦ )with r = 0 . r LC , does not reproduce the data satisfactorily.The results from Sections 4 and 5 indicate that, whilepure or current-modified dipoles are effective models fornon-recycled pulsars (CM10), MSPs are likely to have morecomplicated magnetic geometries. This is not surprising,as the accretion process can significantly distort a pul-sar’s magnetic field (Lai et al. 1999; Payne & Melatos 2004;Lamb et al. 2009). Alternative magnetic configurations in-clude a quadrupole or localized surface anomaly (Lai et al.1999; Long et al. 2008), a force-free field (Spitkovsky 2006;Bai & Spitkovsky 2009), a vacuum-like field (Melatos 1997),or a field distorted by the formation of a polar mag- netic mountain (Payne & Melatos 2004; Vigelius & Melatos2008).We emphasize the utility of the Stokes phase portraitsas a supplementary diagnostic tool for MSPs. The PA swingon its own is especially ambiguous when dealing with non-dipolar fields. Future work will focus on the role played bycircular polarization in Stokes tomography, the longitudinalstructure of vacuum and force-free magnetospheres, and thepolarization signatures of magnetic mountains. These topicswill form the subject of companion papers. ACKNOWLEDGEMENTS c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars APPENDIX: ATLAS OF LOOK-UP TABLES OFSTOKES PHASE PORTRAITS
We present look-up tables for the beam patterns and lin-ear polarization models used in CM10, updated to includeinterpulse and relativistic aberration effects. All figures arefor r = 0 . r LC . Stokes phase portraits and PA swings areshown for a current-modified dipole with(i) a filled core beam with L = I cos θ (Figures 18–21),(ii) a filled core beam with L = I sin θ (Figures 22–25),(iii) a hollow cone with L = I cos θ (Figures 26–29), and(iv) a hollow cone with L = I sin θ (Figures 30–33). c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 18.
Current-modified dipole. Look-up table of Stokes phase portraits in the I - Q plane for filled core beams with degree of linearpolarization L = I cos θ , where θ is the emission point colatitude, and r = 0 . r LC . The panels are organised in landscape mode, in orderof increasing 10 ◦ i ◦ (left–right) and 10 ◦ α ◦ (top–bottom) in intervals of 10 ◦ . I is plotted on the horizontal axis andnormalised by its peak value. Q is plotted on the vertical axis. c (cid:13)000
Current-modified dipole. Look-up table of Stokes phase portraits in the I - Q plane for filled core beams with degree of linearpolarization L = I cos θ , where θ is the emission point colatitude, and r = 0 . r LC . The panels are organised in landscape mode, in orderof increasing 10 ◦ i ◦ (left–right) and 10 ◦ α ◦ (top–bottom) in intervals of 10 ◦ . I is plotted on the horizontal axis andnormalised by its peak value. Q is plotted on the vertical axis. c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 19.
Current-modified dipole. Layout as for Figure 18, but for I - U ( I on the horizontal axis).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 20.
Current-modified dipole. Layout as for Figure 18, but for Q - U ( Q on the horizontal axis).c (cid:13)000
Current-modified dipole. Layout as for Figure 18, but for Q - U ( Q on the horizontal axis).c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 21.
Current-modified dipole. Layout as for Figure 18, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians).c (cid:13) , 1–26 C. T. Y. Chung et al.
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Current-modified dipole. Look-up table of Stokes phase portraits in the I - Q plane for filled core beams with degree of linearpolarization L = I sin θ , where θ is the emission point colatitude, and r = 0 . r LC . The panels are organised in landscape mode, in orderof increasing 10 ◦ i ◦ (left–right) and 10 ◦ α ◦ (top–bottom) in intervals of 10 ◦ . I is plotted on the horizontal axis andnormalised by its peak value. Q is plotted on the vertical axis.c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 23.
Current-modified dipole. Layout as for Figure 22, but for I - U ( I on the horizontal axis).c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 24.
Current-modified dipole. Layout as for Figure 22, but for Q - U ( Q on the horizontal axis).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 25.
Current-modified dipole. Layout as for Figure 22, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians). c (cid:13)000
Current-modified dipole. Layout as for Figure 22, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians). c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 26.
Current-modified dipole. Look-up table of Stokes phase portraits in the I - Q plane for hollow cones with opening angle 25 ◦ and degree of linear polarization L = I cos θ , where θ is the emission point colatitude, and r = 0 . r LC . The panels are organised inlandscape mode, in order of increasing 10 ◦ i ◦ (left–right) and 10 ◦ α ◦ (top–bottom) in intervals of 10 ◦ . I is plotted on thehorizontal axis and normalised by its peak value. Q is plotted on the vertical axis.c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 27.
Current-modified dipole. Layout as for Figure 26, but for I - U ( I on the horizontal axis).c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 28.
Current-modified dipole. Layout as for Figure 26, but for Q - U ( Q on the horizontal axis).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 29.
Current-modified dipole. Layout as for Figure 26, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians). c (cid:13)000
Current-modified dipole. Layout as for Figure 26, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians). c (cid:13)000 , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 30.
Current-modified dipole. Look-up table of Stokes phase portraits in the I - Q plane for hollow cones with opening angle 25 ◦ and degree of linear polarization L = I sin θ , where θ is the emission point colatitude, and r = 0 . r LC . The panels are organised inlandscape mode, in order of increasing 10 ◦ i ◦ (left–right) and 10 ◦ α ◦ (top–bottom) in intervals of 10 ◦ . I is plotted on thehorizontal axis and normalised by its peak value. Q is plotted on the vertical axis.c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 31.
Current-modified dipole. Layout as for Figure 30, but for I - U ( I on the horizontal axis).c (cid:13) , 1–26 tokes tomography of radio pulsar magnetospheres. II. Millisecond pulsars Figure 32.
Current-modified dipole. Layout as for Figure 30, but for Q - U ( Q on the horizontal axis).c (cid:13) , 1–26 C. T. Y. Chung et al.
Figure 33.
Current-modified dipole. Layout as for Figure 30, but for position angle (on the vertical axis in landscape orientation, inunits of radians) versus pulse longitude (on the horizontal axis, in units of 2 π radians). c (cid:13)000